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Extremum Seeking Tracking for Derivative-free Distributed Optimization

Nicola Mimmo, Guido Carnevale, Andrea Testa, Giuseppe Notarstefano

TL;DR

A novel distributed algorithm is proposed that combines a recent gradient tracking policy with an extremum seeking technique to estimate the global descent direction and provides arbitrarily accurate solution estimates through the combination of Lyapunov and averaging analysis approaches with consensus theory.

Abstract

In this paper, we deal with a network of agents that want to cooperatively minimize the sum of local cost functions depending on a common decision variable. We consider the challenging scenario in which objective functions are unknown and agents have only access to local measurements of their local functions. We propose a novel distributed algorithm that combines a recent gradient tracking policy with an extremum seeking technique to estimate the global descent direction. The joint use of these two techniques results in a distributed optimization scheme that provides arbitrarily accurate solution estimates through the combination of Lyapunov and averaging analysis approaches with consensus theory. We perform numerical simulations in a personalized optimization framework to corroborate the theoretical results.

Extremum Seeking Tracking for Derivative-free Distributed Optimization

TL;DR

A novel distributed algorithm is proposed that combines a recent gradient tracking policy with an extremum seeking technique to estimate the global descent direction and provides arbitrarily accurate solution estimates through the combination of Lyapunov and averaging analysis approaches with consensus theory.

Abstract

In this paper, we deal with a network of agents that want to cooperatively minimize the sum of local cost functions depending on a common decision variable. We consider the challenging scenario in which objective functions are unknown and agents have only access to local measurements of their local functions. We propose a novel distributed algorithm that combines a recent gradient tracking policy with an extremum seeking technique to estimate the global descent direction. The joint use of these two techniques results in a distributed optimization scheme that provides arbitrarily accurate solution estimates through the combination of Lyapunov and averaging analysis approaches with consensus theory. We perform numerical simulations in a personalized optimization framework to corroborate the theoretical results.

Paper Structure

This paper contains 18 sections, 4 theorems, 72 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Extremum Seeking Tracking: Algorithm Introduction and Convergence
  4. Extremum Seeking Tracking: Stability Analysis
  5. Coordinate changes and averaged system
  6. Averaged System Analysis
  7. Proof of Theorem \ref{['th:MainResult']}
  8. Numerical Computations on Distributed personalized optimization
  9. Monte Carlo simulations for large-scale problems
  10. Monte Carlo simulations varying number of agents
  11. Monte Carlo simulations varying the size $n$
  12. Monte Carlo simulations varying the parameter $\delta$
  13. Monte Carlo simulations for comparisons
  14. Monte Carlo simulations in stochastic scenarios
  15. Conclusions
  16. ...and 3 more sections

Key Result

Theorem 1

Consider eq:GTA and let Assumptions ass:connectivity, ass:strong_convexity, and ass:lipschitz hold. Then, for any $r, \bar{\rho} > 0$, there exist $\gamma^\star, \delta^\star, k_1 > 0$, $\epsilon \in (0,\bar{\rho}/2)$, and $k_2 \ge (\bar{\rho}/2 - \epsilon)$ such that, for any $\gamma \in (0,\gamma^ for all $i \in \{1,\ldots,N\}$ and $t \ge t^\star :=\! -\frac{1}{\gamma k_1}\ln((\bar{\rho}/2 \! -

Figures (8)

  • Figure 1: Block scheme of the proposed Extremum Seeking Tracking algorithm in the $(x,z)$ coordinates.
  • Figure 2: Monte Carlo simulations for large scale problems: mean and $1$-standard deviation band of cost error (left) and variable error (right).
  • Figure 3: Monte Carlo simulations for a varying number of agents: mean and $1$-standard deviation band of cost error (left) and variable error (right).
  • Figure 4: Monte Carlo simulations varying $n$: mean and $1$-standard deviation band of cost error (left) and variable error (right).
  • Figure 5: Evolution of the agents' estimates $w_i^t$ in coordinate error with respect to the optimal solution $x^\star$.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Lemma 1: Gradient estimation
  • Lemma 2
  • Remark 3
  • Lemma 3
  • Remark 4